Investigate the improper integral for conditional and absolute convergence at different values of the parameter α $$\int_{0}^{1}\frac{1-x}{x^{2}\sin(x^{\alpha })}dx$$
As I understand I should consider the cases α=0, α>0, α<0. I may use equivalence in the first two cases. But what else I can use here and how? In the latter case,I make a replacement $t= x^{\alpha }$ but in the neighborhood of the zeros of the sine will be bad because there would be a bunch of zeroes of sine and all are not integrable Can anyone please help me to solve it?